Shortest Paths Help Solve Geometric Optimization Problems in Planar Regions

Melissaratos, Elefterios A.; Souvaine, Diane L.

doi:10.1137/0221038

Cited by 36 publications

(52 citation statements)

References 29 publications

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“…This can be done in O(n) time [3], [6], [7]. This algorithmic question was also studied without the convexity assumption (finding the maximum area triangle contained in a simple n-gon), where it becomes much harder [17]. The perimeter result also holds for any strictly convex norm, since we use only the triangle inequality.…”

Section: Theorem 2 a Set Of N Points In The Plane Determines At Mostmentioning

confidence: 95%

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Braβ

Rote

Swanepoel³

2001

Discrete Comput Geom

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Section: Theorem 2 a Set Of N Points In The Plane Determines At Mostmentioning

confidence: 95%

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Braβ

Rote

Swanepoel³

2001

Discrete Comput Geom

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“…A (simple) splinegon S is a simple region formed by replacing each edge e i of a simple polygon P by a curved edge e i joining the endpoints of e i such that the area bounded by the curve e i and the line segment e i is convex (see Figure 1). The vertices of S are the vertices of P. As in Dobkin and Souvaine [1990], Dobkin et al [1988], and Melissaratos and Souvaine [1992], we assume that the combinatorial complexity of each splinegon edge is O(1), and primitive operations on a splinegon edge can each be performed in O(1) time, such as computing the intersections of a splinegon edge with a line, computing the tangents (if any) between two splinegon edges, finding the tangents between a point and a splinegon edge, computing the distance between two points along a splinegon edge, etc.…”

Section: The Geometric Setting and Our Resultsmentioning

confidence: 99%

“…As in Dobkin and Souvaine [1990], Dobkin et al [1988], and Melissaratos and Souvaine [1992], we use splinegons to model planar curved objects. A (simple) splinegon S is a simple region formed by replacing each edge e i of a simple polygon P by a curved edge e i joining the endpoints of e i such that the area bounded by the curve e i and the line segment e i is convex (see Figure 1).…”

Section: The Geometric Setting and Our Resultsmentioning

confidence: 99%

Computing Shortest Paths among Curved Obstacles in the Plane

Chen

Wang

2015

ACM Trans. Algorithms

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A fundamental problem in computational geometry is to compute an obstacle-avoiding Euclidean shortest path between two points in the plane. The case of this problem on polygonal obstacles is well studied. In this article, we consider the problem version on curved obstacles, which are commonly modeled as splinegons. A splinegon can be viewed as replacing each edge of a polygon by a convex curved edge (polygons are special splinegons), and the combinatorial complexity of each curved edge is assumed to be O(1). Given in the plane two points s and t and a set S of h pairwise disjoint splinegons with a total of n vertices, after a bounded degree decomposition of S is obtained, we compute a shortest s-to-t path avoiding the splinegonswhere k is a parameter sensitive to the geometric structures of the input and is upper bounded by O(h 2 ). The bounded degree decomposition of S, which is similar to the triangulation of the polygonal domains, can be computed in O(n log n) time or O(n + h log 1+ h) time for any > 0. In particular, when all splinegons are convex, the decomposition can be computed in O(n + h log h) time and k is linear to the number of common tangents in the free space (called "free common tangents") among the splinegons. Our techniques also improve several previous results: (1) For the polygon case (i.e., when all splinegons are polygons), the shortest path problem was previously solved in O(n log n) time, or in O(n+ h 2 log n) time. Thus, our algorithm improves the O(n+ h 2 log n) time result, and is faster than the O(n log n) time solution for sufficiently small h, for example, h = o( n log n). (2) Our techniques produce an optimal output-sensitive algorithm for a basic visibility problem of computing all free common tangents among h pairwise disjoint convex splinegons with a total of n vertices. Our algorithm runs in O(n+h log h+k) time and O(n) working space, where k is the number of all free common tangents. Note that k = O(h 2 ). Even for the special case where all splinegons are convex polygons, the previously best algorithm for this visibility problem takes O(n + h 2 log n) time. (3) We improve the previous work for computing the shortest path between two points among convex pseudodisks of O(1) complexity each.In addition, a by-product of our techniques is an optimal O(n + h log h) time and O(n) space algorithm for computing the Voronoi diagram of a set of h pairwise disjoint convex splinegons with a total of n vertices.

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“…2, middle); build "conservative" obstacles [39] for (the centerline of) the kth thick path by inflating each hole H by d k (H) (Fig. 2, right); find the shortest S-T path in the free space (which is a simple splinegon) using bounded-degree splinegonal decomposition (analog of triangulation) of the free space [14,36]. The final result is:…”

Section: Shortest Homotopic Routingmentioning

confidence: 99%

“…We compute schrp( π, w) path-by-path using tools from [14,24,36,39]. We determine "depths" of the holes w.r.t.…”

Section: Shortest Homotopic Routingmentioning

confidence: 99%

Optimal Geometric Flows via Dual Programs

Eriksson-Bique

Polishchuk²,

Sysikaski

2014

Proceedings of the Thirtieth Annual Symposium on Computational Geometry

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Considering potentials in the dual of a planar network has proved to be a powerful tool for computing planar maximum flows. In this paper we explore the use of potentials for giving algorithmic and combinatorial results on continuous flows in geometric domains -a (far going) generalization of discrete flows in unit-capacity planar networks.A continuous flow in a polygonal domain is a divergencefree vector field whose magnitude at every point is bounded by a given constant -the domain's permeability. The flow enters the domain through a source edge on the outer boundary of the domain, and leaves through a sink edge on the boundary. The value of the flow is the total flow amount that comes in through the source (the same amount comes out of the sink, due to the vanishing divergence). The cost of the flow is the total length of its streamlines. The flow is called monotone if its streamlines are x-monotone curves.Our main result is an algorithm to find (an arbitrarily close approximation to) the minimum-cost monotone flow in a polygonal domain, by formulating the problem as a convex program with an interesting choice of variables: one variable per hole in the domain. Our approach is based on the following flow of ideas: flow is the gradient of a potential function; a potential function can be extended from free space to holes so that it is constant over each hole; instead of the potential function (a value per point in the domain) we can thus speak of a potential vector (a value per hole); a potential uniquely (up to homotopic equivalence) defines "thick" paths in the domain; the paths define a flow. We show that the flow cost is convex in the potential; this allows us to employ the ellipsoid method to find the mincost flow, using holes potentials as variables. At each ellipsoid iteration the separation oracle is implemented by feasibility checks in the "critical graph" of the domain -as we prove, the graph defines linear constraints on the potentials.Using potentials and critical graphs we also prove MaxFlow/ MinCut theorems for geometric flows -both for monotone and for general ones. Formulating and proving the monotone Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from Permissions@acm.org.

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Shortest Paths Help Solve Geometric Optimization Problems in Planar Regions

Cited by 36 publications

References 29 publications

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Triangles of Extremal Area or Perimeter in a Finite Planar Point Set

Computing Shortest Paths among Curved Obstacles in the Plane

Optimal Geometric Flows via Dual Programs

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